To reiterate, a stochastic model of population dynamics that incorporates individual life histories is more appropriate for long-lived species with overlapping generations than are deterministic models. It is particularly appropriate for small population dynamics or, in the case of larger populations, those in highly variable environments.
Lilian Wu and Daniel Botkin presented in 1980 the first stochastic model designed specifically for studying the dynamics of African elephants. Their mathematical model provided the analytical framework for female life histories. Using a matrix approach, the model provided methods for calculating the relationship between change in population size and age-specific fecundity and mortality, the average number of female offspring produced by a female in her lifetime, the probability of extinction of a population, and if the population did not become extinct, composition of the population after a certain period of time.
Wu and Botkin then considered the age distribution of female elephants obtained from a 1972 cull in Wankie (Hwange) National Park, Zimbabwe. The age distribution of this sample was irregular, suggesting that it could be due to one or more of the following factors: (1) chance errors in sampling individuals from a stable aged structure; (2) some unknown past event had disrupted the population, which was now moving toward a stable age distribution; or (3) variations in rainfall could have caused this nonstationary population age structure. Using data on birth and death probabilities from the East African study by Richard Laws, they showed from their formulation that the irregular age distribution of the Wankie sample could not possibly represent a stable aged population. They concluded that variations in rainfall were the most likely cause of time-varying birth and death processes and of irregular age profiles.
Since its publication in 1980, the Wu-Botkin stochastic model has never actually been applied to any elephant population except for a test of the Wan-kie age structure. The complex mathematical formulation of the analytical solutions would have inhibited biologists from even attempting to use the model. The spread of personal computers made simulations of stochastic dynamics more attractive to use. I now discuss the use of simulations in determining probability of extinction and population viability in Asian and in African elephants.
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